A gyroscope is a sensor that measures orientation. This is often referred to as angular momentum and this can be measured in three dimentions perpendicular to each other. The three axis are often referred to as the pitch roll and yaw axis derived from the aviation industru. The angular momentum is directional and in aircraft terminology the pitch is positive when the nose of the aircraft goes up and negative when it goes down. Similarly when the left wing goes down the roll is positive and negative when the right wing goes down. Yaw is when there is rotation in the earth plain and is positive with clockwise rotation. The sensors which you can use for interfacing to the arduino are MEMS sensors. These micro electro-mechanical systems (MEMS) sensors output the angular momentum in a way that microprocessors can understand. This can be I2C or as an analog signal. The datasheets of the different components will help you to convert these values to degrees per second or whatever real world units you are using. There are two gyroscopes i have experimented with which are the ITD300 which was one of the early ones and the ITG3200. Gyroscopes are accurate in the short term but have a tendency to drift over time. This means that you can not rely on their output over a longer period of time. Practically you can compensate this drifting with the output from an accelerometer. Accelerometers have a jitter so there short term readings are not so reliable. In the market there are sophisticated filters which use the output from both types of sensors and merge them into a single reliable output. These are for example DCM or Kalman filters and details of how they work can be found on the connecting pages.
Links to the gyroscopes
In these articles you can also find a method of how to make an artificial horizon respond to movements of the sensors. Vey cool !!!